Why j for imaginary unit? | Hacker News Reader

Why j for imaginary unit?

johndcook.com

34 pointsedwintorok24 comments

But this is only a problem if you use i, j, and k as unit vectors. I found $\hat e_{x,y,z}$ or even $\hat{e}_{1,2,3…}$ etc. to be the usual convention in physics, as it generalises much nicer to more dimensions.

I agree. I was just about to write a similar comment.

Before vector notation became popular in the last 1800s, people used quaternions. The three independent square roots of -1 in quaternion notation are i, j, and k. Therefore the use of j or (in need) k as a square root of -1 seems natural to me.

My understanding is that this history is why people in physics - even today - frequently use i, j, and k as the names of the three spatial unit vectors. (The real numbers, of course, represented time.) However I've not personally looked into any of this history, so you should treat my understanding more as hearsay rather than informed comment.

People still use quaternions. They are an excellent method of representing spatial rotations, since they avoid Gimbal lock (and similar phenomena caused singularities in polar coordinates).

https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

Addendum: frequently used in Aerospace, Navigation, and Motion Tracking applications.

And computer games!

I know that quaternions have been revived. But for the most part they are not heavily used in physics, and therefore their impact on notation is based on past usage, not current usage.

I actually see that as an argument against using j. The meanings of a unit vector in real space and a dimensional unit in imaginary space are different concepts even if it's common to represent both in 2-dimensions plots. It seems to me that this would only cause confusion/ambiguity. When I look at a plot I want to know what it's representing as soon as possible, and the way the axes are labeled is part of what helps me.

Exactly what I thought. I studied EE in undergrad so I used this sort of notation all the time, and my thought upon reading the article was that I'd be rather confused upon looking at the graph and would struggle to figure out what it was even representing.

Yes, ultimately we are bound by tradition, since changing over to any system is too expensive and confusing.

Path dependence and local optima.

See also: English vs Metric units.

agreed. i think this convenience is confusing. also would make it hard to explain that a two-dimensional complex vector contains two complex numbers, not one.

Want to know what's worse? Spherical coordinates.

http://en.wikipedia.org/wiki/Spherical_coordinates#Conventio...

You need to remember two different sets of conversions from cartesian to spherical coordinates and two sets of differential elements -and of course, you need to remember which is assumed by which audience.

Speaking as someone just graduating high school in India, this sort of unintuitive convention discrepancy often flummoxes people at the 9th grade level, causing them to drop out of Physics altogether. Once you drop Physics, it's essentially impossible to get onto the Engineering or CS tracks in India, and U.S. universities are often too expensive.

I've occasionally considered that most of my "engineering training" is all about working around human weirdness like this. On the bright side someone dropping out because of i vs j won't have to suffer through conventional current vs electron current, aka the same thing as the right hand rule vs the left hand ("FBI") rule. Even worse some call the RHR the generator rule and the LHR the motor rule. This is a cousin of people confusing right and left hand circular polarization of antennas vs waves etc.

Somewhat more practically old fashioned electrolytic capacitors pretty much always mark the negative lead with a black mark or stripe in excess of 99.99% of the time (yeah I've seen at least 10K and never seen an exception yet). However I have to crack open a datasheet every time for every tantalum because a "large fraction" of them mark the positive instead of the negative... lovely.

This is before we start playing games with some people talking about RMS voltages, some peak, some pk-pk.

I believe this is why many engineers don't care about imperial vs metric. Its just another example of standards being so desirable we'd like to have as many as possible. Whats one more?

Perhaps my comment will be considered harsh, but my view has long been that if you are hung up on the notation, you miss the physics, you miss the science.

There is no such as "an intuitive convention". Notation is simply that: Convention. If someone cannot grasp this, they lack either or both of the intellectual maturity and the doggedness necessary to understand the science.

If teachers failed to make the point that notation is simply convention, chosen for largely historical reasons, then they have not helped the situation - but fundamentally it is up to the student to see past, to see through the conventions and to perceive what the notation is saying.

Why does "W" represent the "wa" sound? Convention. Do you see the individual letters or do you read the words, the sentences?

Arguing whether you should use i or j for square root -1 is like arguing whether you should o or u for a particular vowel sound.

> If someone cannot grasp this, they lack either or both of the intellectual maturity

Yes, that's entirely the point! >> flummoxes people at the 9th grade level,

> There is no such as "an intuitive convention".

consistency is intuitive. inconsistency is unintuitive. That's not the same as a hypothetical complaint that a certain convention is unintuitive.

Even if we set out to build an entirely consistent set of notation for use in all of physics, maths and engineering today, we would face obvious problems, mostly that there are only about 100 different symbols (Latin + Greek + some extras) with maybe a handful of modifiers (primes, dots, tildes…and no, combining them is not good) and typefaces (difficult to replicate in handwriting).

So if we wanted to name every quantity and every concept possibly encountered by physics/maths/engineering today consistently, we already would have a hard time fitting everything in. Add to this that you also need a ‘working space’ of free, unencumbered symbols, a space of symbols free to use in the future[0] and the possibility that seemingly unrelated concepts today might well be linked in a few years time, and you will come to the conclusion that there is no way we could possibly build a consistent set of notation.

Really, be happy with what we have at the moment, choose a convention suitable to your current field of work[1] and everything will be fine.

[0] Some people started using Arab letters for new mathematical functions. Try googling for that.

[1] Or even invent the n-th one if you work at the edge of two existing fields or recognise that you can get stuff done more easily if you choose other symbols.

Relevant xkcd: http://xkcd.com/927/

Although I largely agree with your point that gettin hung up on notation is missing the forest for the trees, you should not entirely dismiss the art of good notation. For example, Leibniz differentials make the chain rule seem "obvious" due to its analogy with standard fractions.

But, I agree that people who get hung up on the same symbol having different unrelated meanings or different symbols having the same meaning will have a hard time learning the physics beneath the symbols.

> this sort of unintuitive convention discrepancy often flummoxes people at the 9th grade level, causing them to drop out of Physics altogether

that's a rather weirdly extreme reaction to a notation, imho. it is just a 'convention', accept and move on. is there something more than meets the 'i' ?

I'd think tau vs. pi does far more damage to our kids learning. Radians would be so much easier for highschoolers to grasp if it was 1tau per revolution instead of 2pi.

An interesting side-point -- Python supports imaginary numbers and uses j:

      In [2]: a=5-4j

      In [3]: b=1+2j

      In [4]: a+b
      Out[4]: (6-2j)

Another interesting side-point -- MATLAB supports imaginary numbers... unless of course you redefine i to be something else. Then it doesn't.

Fortunately, Mathematica makes that somewhat more difficult (and has a strong convention of inbuilt ./. user-defined variables in the form of upper/lowercase).

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